Kollineationsgruppen kompakter, vier-dimensionaler Ebenen

Salzmann, Helmut

doi:10.1007/bf01109833

Cited by 69 publications

(19 citation statements)

References 18 publications

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“…According to [1,Lemma 5] the trans-lation plane ~ is Desarguesian or W fixes L~. If dim W >t 9, then ~ is Desarguesian according to [12] and [13]. So let us assume that dim W = 8 and that L~o is fixed under W. In this case E is a subgroup of W of codimension 1.…”

Section: Proposition Assume That £ Fixes a Point P Then P Is The Unmentioning

confidence: 98%

On 4-dimensional Minkowski planes with 7-dimensional automorphism groups

Steinke

1993

Geom Dedicata

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ABSTRACT. This paper concerns 4-dimensional (topological locally compact connected) Minkowski planes that admit a 7-dimensional automorphism group E. It is shown that such a plane is either classical or has a distinguished point that is fixed by the connected component of E and that the derived affine plane at this point is a 4-dimensional translation plane with a 7-dimensional collineation group.A Minkowski plane J//= (P, ~, { [I +, II -}) consists of a set of points P, a set of at least two circles ~ (considered as subsets of P) and two equivalence relations I[ + and I[ -on P (parallelisms) such that three pairwise non-parallel points (that is, neither (+)-parallel nor (-)-parallel) can be joined by a unique circle, such that the circles which touch a fixed circle K at p E K partition P\lPl (here [p[ = I pt + u [pl-denotes the union of the two parallel classes of p), such that each parallel class meets each circle in a unique point (parallel projection), such that each (+)-parallel class and each (-)-parallel class intersect in a unique point, and such that there is a circle that contains at least three points (compare [15]). A topological Minkowski plane is a Minkowski plane in which the point set P and the set of circles j~r carry topologies such that the geometric operations of joining, touching, the parallel projections, intersecting parallel classes of different type, and intersecting circles are continuous operations on their domains of definition (see 1-15]). A topological Minkowski plane is called (locally) compact, connected, or finite-dimensional if the point space has the respective topological property. For brevity, a locally compact connected finite-dimensional topological Minkowski plane will be called a finite-dimensional Minkowski plane. According to 1-9, 2.3] a finite-dimensional Minkowski plane can only be of dimension 2 or 4. In these cases the automorphism group of J/is a Lie group with respect to the compact-open topology of dimension at most 6 and 12, respectively, see [16]. The classical model of a 2-or 4-dimensional Minkowski plane is obtained as the geometry of non-trivial plane sections of a ruled quadric in the real or complex projective 3-dimensional space respectively. In these cases the topologies on the point set and the set of circles are induced from the surrounding projective 3-space (the set of planes in the projective 3-space carries a natural topology which can be obtained by duality from the topology of the point set in the 3-space).

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Section: Proposition Assume That £ Fixes a Point P Then P Is The Unmentioning

confidence: 98%

On 4-dimensional Minkowski planes with 7-dimensional automorphism groups

Steinke

1993

Geom Dedicata

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“…b) If ~ --(Q, ~) is a homogeneous Baer subplane, then F 1 acts faithfully on Q by [12], 4.1, and contains the elliptic motion group S~ SO3 of~ ~ PER by [11], 5.1. c) Conversely, consider a group SO3 ~ q~ ~< F. We show that contains both a homogeneous oval and a homogeneous Baer subplane.…”

Section: Theorem 2 the Following Assertions About A 4-dimensional Comentioning

confidence: 99%

“…Because ~b is connected, J consists of reflections ( [12], 4.1). The corresponding centers form an orbit Q of q~.…”

Section: Theorem 2 the Following Assertions About A 4-dimensional Comentioning

confidence: 99%

Compact projective planes with homogeneous ovals

Löwen

1984

Monatshefte f�r Mathematik

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Abstract. Closed ovals exist only in 2-or 4-dimensional compact projective planes.We show that a plane of dimension 2 or 4 contains a homogeneous closed oval iff the automorphism group contains SO 2 or SO3, respectively. In 4-dimensional planes, existence of a homogeneous oval and existence of a homogeneous Baer subplane are equivalent. We determine the possible full automorphism groups of planes containing homogeneous ovals except for two possibilities in dimension 4 whose existence remains uncertain.The group PSO3 C of automorphisms of the complex projective plane fixes an oval (in fact, a conic), and PSL3 E fixes a Baer subsplane. So their intersection (P)SO3 E fixes both. Our aim is to show that this is a general phenomenon with SO3 E-actions on 4-dimensional compact planes. Our Theorem 5 suggests that non-desarguesian examples will be hard to construct.

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“…We prove the following theorem: Let 7 ~ be an 8-dimensional compact topological projective plane. If the connected component A of its automorphism group has dimension at least 12, then A is a Lie group.Mathematics Subject Classification (1991): 51H 10.For any compact connected topological projective plane 7', whose point space P has dimension at most 4, we know that P is a manifold (in fact P ~ P2~ or P ~ P2C respectively) and that the automorphism group of 7" is a Lie group (see [13]). If the point space has dimension greater than 4, it is not known whether or not it is a manifold.…”

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confidence: 99%

Large automorphism groups of 8-dimensional projective planes are Lie groups

Priwitzer

1994

Geom Dedicata

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Abstract. We prove the following theorem: Let 7 ~ be an 8-dimensional compact topological projective plane. If the connected component A of its automorphism group has dimension at least 12, then A is a Lie group.Mathematics Subject Classification (1991): 51H 10.For any compact connected topological projective plane 7', whose point space P has dimension at most 4, we know that P is a manifold (in fact P ~ P2~ or P ~ P2C respectively) and that the automorphism group of 7" is a Lie group (see [13]). If the point space has dimension greater than 4, it is not known whether or not it is a manifold. The automorphism group of the projective plane is always a locally compact transformation group of P, but it is still an open question if it is always a Lie group. For the most interesting cases, however, i.e. for automorphism groups of sufficiently large dimension, we can show that they are Lie groups. Thus, the purpose of this paper is to prove the following theorem: THEOREM. The connected component A of the automorphism group of an 8-dimensional compact connected projective plane is a Lie group if dim A > 12.Remark. An analogous result for 16-dimensional planes was published by LOwen and Salzmann [8]. They proved that an automorphism group is a Lie group if its dimension is at least 36. Using methods of the present paper, this bound can be improved considerably: 28 is already sufficient. PreliminariesThe notation in this paper is as follows: 79 will always denote a compact connected 8-dimensional projective plane with point space P and line space/2. The connected component of its automorphism group is called A, and A _< A is the stabilizer of a nondegenerate quadrangle. For X C P we denote by Ax the stabilizer of X, and by A[x ] its pointwise stabilizer. The set of fixed points of an element E A is denoted by Fs. For the dimension of a coset space we write dim A/F = dimA-dimP = A.F.

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Kollineationsgruppen kompakter, vier-dimensionaler Ebenen

Cited by 69 publications

References 18 publications

On 4-dimensional Minkowski planes with 7-dimensional automorphism groups

On 4-dimensional Minkowski planes with 7-dimensional automorphism groups

Compact projective planes with homogeneous ovals

Large automorphism groups of 8-dimensional projective planes are Lie groups

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